Abstract
Fourier series are ideal for analyzing periodic signals, since the harmonic modes used in the expansions are themselves periodic. The Fourier integral transform is a far less natural tool because it uses periodic functions to expand nonperiodic signals. Two possible substitutes are the windowed Fourier transform and the wavelet transform. In this chapter we introduce the Fourier transform, the windowed Fourier transform and wavelets (Debnath and Mikusinski [16], Chui [13], Kaiser [30], Chan [12]). We show how the windowed Fourier transform and the wavelet transform can be used to give information about signals simultaneously in the time domain and the frequency domain. We then derive the counterpart of the inverse Fourier transform, which allows us to reconstruct a signal from its windowed Fourier transform. We find a necessary and sufficient condition that an otherwise arbitrary function of time and frequency must satisfy in order to be the windowed Fourier transform of a time signal with respect to a given window and introduce a method of processing signals simultaneously in time and frequency.
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© 1998 Springer Science+Business Media Dordrecht
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Steeb, WH. (1998). Fourier Transform and Wavelets. In: Hilbert Spaces, Wavelets, Generalised Functions and Modern Quantum Mechanics. Mathematics and Its Applications, vol 451. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5332-4_2
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DOI: https://doi.org/10.1007/978-94-011-5332-4_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6241-1
Online ISBN: 978-94-011-5332-4
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